Computer Organization Computer Organization Review: Combinational circuits Week 1 Lecture Notes Adapted by Dr. Adel Ammar

Computer Organization Combinational vs. sequential digital circuits v

Computer Organization Part 1: Design

Computer Organization 4 Overview vDesign digital circuit from specification vDigital inputs and outputs known ðNeed to determine logic that can transform data vStart in truth table form vCreate K-map for each output based on function of inputs vDetermine minimized sum-of-product representation vDraw circuit diagram

Computer Organization 5 Design Procedure vBoolean algebra can be used to simplify expressions, but not obvious: ðhow to proceed at each step, or ðif solution reached is minimal. vThere are five ways to represent a function: ðBoolean expression ðtruth table ðlogic circuit ðminterms/maxterms ðKarnaugh map

Computer Organization 6 Combinational logic design vUse multiple representations of logic functions vUse graphical representation to assist in simplification of function. vUse concept of “don’t care” conditions. vExample ðencoding BCD to seven segment display.

Computer Organization 7 BCD to Seven Segment Display vUsed to display binary coded decimal (BCD) numbers using seven illuminated segments. vBCD uses 0’s and 1’s to represent decimal digits Need four bits to represent required 10 digits. vBinary coded decimal (BCD) represents each decimal digit with four bits a b c g e d f

Computer Organization 8 BCD to seven segment display vList the segments that should be illuminated for each digit. 0a,b,c,d,e,f 1b,c 2a,b,d,e,g 3a,b,c,d,g 4b,c,f,g 5a,c,d,f,g 6a,c,d,e,f,g 7a,b,c 8a,b,c,d,e,f,g 9a,b,c,d,f,g a b c g e d f

Computer Organization 9 BCD to seven segment display vDerive the truth table for the circuit. vEach output column in one circuit edcbazyxwDec  InputsOutputs

Computer Organization 10 BCD to seven segment display vFind minimal sum-of-products representation for each output yz wx For segment “a” : Note: Have only filled in ten squares, corresponding to the ten numerical digits we wish to represent.

Computer Organization 11 Don’t care conditions (BCD display)... vFill in don’t cares for undefined outputs. ðNote that these combinations of inputs should never happen. vLeads to a reduced implementation XX XXXX yz wx For segment “a” : Put in “X” (don’t care), and interpret as either 1 or 0 as desired ….

Computer Organization 12 Don’t care conditions (BCD display)... vCircle biggest group of 1’s and Don’t Cares. vLeads to a reduced implementation For segment “a” : XX XXXX yz wx

Computer Organization 13 Don’t care conditions (BCD display) vCircle biggest group of 1’s and Don’t Cares. vLeads to a reduced implementation For segment “a” : XX XXXX yz wx

Computer Organization 14 Don’t care conditions (BCD display)... vCircle biggest group of 1’s and Don’t Cares. vAll 1’s should be covered by at least one implicant For segment “a” : XX XXXX yz wx XX XXXX yz wx

Computer Organization 15 Don’t care conditions (BCD display)... v Put all the terms together vGenerate the circuit For segment “a” : XX XXXX yz wx

Computer Organization 16 BCD to seven segment display vDerive the truth table for the circuit. vEach output column in one circuit edcbazyxwDec  InputsOutputs

Computer Organization 17 BCD to seven segment display vFind minimal sum-of-products representation for each output yz wx For segment “b” : See if you complete this example.

Computer Organization 18 Summary vNeed to formulate circuits from problem description ðDetermine number of inputs and outputs ðDetermine truth table format ðDetermine K-map ðDetermine minimal SOP vThere may be multiple outputs per design ðSolve each output separately vCurrent approach doesn’t have memory.

Computer Organization Part 2: Adder and substractor

Computer Organization 20 vAddition and subtraction of binary data is fundamental ðNeed to determine hardware implementation vRepresent inputs and outputs ðInputs: single bit values, carry in ðOutputs: Sum, Carry vHardware features ðCreate a single-bit adder and chain together vSame hardware can be used for addition and subtraction with minor changes vDealing with overflow ðWhat happens if numbers are too big? Overview

Computer Organization 3.5 Half Adder 21 vCombinational logic circuits give us many useful devices. vOne of the simplest is the half adder, which finds the sum of two bits.

Computer Organization Half Adder 22 vAs we see, the sum can be found using the XOR operation and the carry using the AND operation.

Computer Organization Full adder 23 vWe can change our half adder into to a full adder by including gates for processing the carry bit.

Computer Organization 3.5 Full adder 24 vHow can we change the half adder shown below to make it a full adder?

Computer Organization Full adder 25 vHere’s our completed full adder.

Computer Organization 26 Full Adder vFull adder made of several half adders Half-adder

Computer Organization 27 Full Adder vHardware repetition simplifies hardware design ðA full adder can be made from two half adders (plus an OR gate).

Computer Organization 28 Full Adder vPutting it all together ðSingle-bit full adder ðCommon piece of computer hardware Block Diagram

Computer Organization 29 4-Bit Adder vChain single-bit adders together. C A B S

Computer Organization 30 Negative Numbers – 2’s Complement. vSubtracting a number is the same as: 1.Perform 2’s complement 2.Perform addition vIf we can augment adder with 2’s complement hardware? 1 10 = = = FF 16 = = = = =

Computer Organization 31 4-bit Subtractor: E = 1 +1 Add A to B’ (one’s complement) plus 1 That is, add A to two’s complement of B D = A - B

Computer Organization 32 Adder- Subtractor Circuit

Computer Organization 33 Overflow in two’s complement addition vDefinition: When two values of the same signs are added: ðResult won’t fit in the number of bits provided ðResult has the opposite sign.

Computer Organization OFL Addition cases and overflow

Computer Organization 35 Summary: Adder-Substractor vAddition and subtraction are fundamental to computer systems vKey – create a single bit adder/subtractor ðChain the single-bit hardware together to create bigger designs vThe approach is called ripple-carry addition ðCan be slow for large designs vOverflow is an important issue for computers ðProcessors often have hardware to detect overflow